2.3.35 and 2.3.49

2.3.35 and 2.3.49 are two subsets of the 2.3.5.7 (septimal) group that involve many ratios further from 12edo than 2.3.7. The 35th harmonic (5*7) perfectly combines the deviations of the 5th and 7th harmonics to be about as far from 12edo as possible, whereas the 49th harmonic (7*7) overshoots slightly.

Intervals

2.3.35

2.3.35 intervals are the difference between a 5-over and 7-under interval, or vice versa.

36/35 septimal quarter tone (flat of 28/27 by 245/243)

35/32 septimal neutral second

81/70 septimal semifourth

The best scales to use 2.3.5.7 with significant emphasis on 35 is a rank-3 scale consisting of either Meantone or Archy diatonic with a 35/32 offset. One small edo to do something like this is 25, which uses the notes of 5edo to approximate Archy.

2.3.5.49

2.3.49 intervals are the difference between a 7-over and 7-under interval. 49/5 is further from 12edo.

54/49 larger neutral second

49/45 smaller neutral second

49/40 neutral third

2.3.5.49 tripentatonic 5L2m5s3a (blackdye A Aeolian with added chain of 5/49 offset)

81/80

54/49

9/8

6/5

64/49

4/3

27/20

72/49

3/2

8/5

256/147

16/9

9/5

96/49

2/1

<adv>Tempering out 245/243 equates the two smallest steps, leaving 5L2m8s. Tempering out 1029/1024 equates 256/147 to 7/4. Tempering out both commas leads to Rodan temperament (2.3.5.7 41 & 46), an extension of Slendric which also notably tempers out 5120/5103.</adv>

Over-49 Chain

3^	/49	*245/243	*1029/1024
-1	256/147	1280/729	7/4
0	64/49	320/243	21/16
1	96/49	160/81	63/32
2	72/49	40/27	189/128
3	54/49	10/9	567/512
4	81/49	5/3	1701/1024
5	243/196	5/4	5103/4096